Question: The sum of two angles is $78^\circ$. Angle 2 is $122^\circ$ smaller than $4$ times angle 1. What are the measures of the two angles in degrees?
Let $x$ equal the measure of angle 1 and $y$ equal the measure of angle 2. The system of equations is then: ${x+y = 78}$ ${y = 4x-122}$ Since we already have solved for $y$ in terms of $x$ , we can use substitution to solve for $x$ and $y$ Substitute ${4x-122}$ for $y$ in the first equation. ${x + }{(4x-122)}{= 78}$ Simplify and solve for $x$ $ x+4x - 122 = 78 $ $ 5x-122 = 78 $ $ 5x = 200 $ $ x = \dfrac{200}{5} $ ${x = 40}$ Now that you know ${x = 40}$ , plug it back into $ {y = 4x-122}$ to find $y$ ${y = 4}{(40)}{ - 122}$ $y = 160 - 122$ ${y = 38}$ You can also plug ${x = 40}$ into $ {x+y = 78}$ and get the same answer for $y$ ${(40)}{ + y = 78}$ ${y = 38}$ The measure of angle 1 is $40^\circ$ and the measure of angle 2 is $38^\circ$.